874 research outputs found
Modalities in homotopy type theory
Univalent homotopy type theory (HoTT) may be seen as a language for the
category of -groupoids. It is being developed as a new foundation for
mathematics and as an internal language for (elementary) higher toposes. We
develop the theory of factorization systems, reflective subuniverses, and
modalities in homotopy type theory, including their construction using a
"localization" higher inductive type. This produces in particular the
(-connected, -truncated) factorization system as well as internal
presentations of subtoposes, through lex modalities. We also develop the
semantics of these constructions
Accessible higher modalities
Treballs finals del Mà ster en Matemà tica Avançada, Facultat de Matemà tiques, Universitat de Barcelona: Curs: 2022-2023. Director: Carles Casacuberta[en] The main goal of this master’s thesis is to study localizations in the setting of homotopy type theory. In particular the special case of modalities, an object resembling the namesake notion in classical logic that allows to qualify statements to express things as possibility or necessity.
In the first chapter we introduce the syntax of homotopy type theory and some motivating examples for the study of modalities. Next, we study localizations and modalities as described in [1; 2]. We focus on the fact that accessible modalities correspond to nullifications and work on the existence of non-accessible modalities.
Finally we explore the use of Agda (a programming language for automatic proof verification) towards implementing results about factorization systems and modalities
Goodwillie's Calculus of Functors and Higher Topos Theory
We develop an approach to Goodwillie's calculus of functors using the
techniques of higher topos theory. Central to our method is the introduction of
the notion of fiberwise orthogonality, a strengthening of ordinary
orthogonality which allows us to give a number of useful characterizations of
the class of -excisive maps. We use these results to show that the pushout
product of a -equivalence with a -equivalence is a
-equivalence. Then, building on our previous work, we prove a
Blakers-Massey type theorem for the Goodwillie tower. We show how to use the
resulting techniques to rederive some foundational theorems in the subject,
such as delooping of homogeneous functors.Comment: 40 pages, (a slightly modified version of) this paper is accepted for
publication by the Journal of Topolog
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